# QG five principles: superpos. locality diff-inv. cross-sym. Lorentz-inv.

1. Aug 7, 2010

### marcus

== http://arxiv.org/abs/1004.1780 ==

III. TRANSITION AMPLITUDES

In a general covariant quantum theory, the dynamics can be given by associating an amplitude to each boundary state [32, 33]. Therefore, the dynamics is given by a linear functional W on H. The modulus square

P(ψ) = |⟨W|ψ⟩|2

is the probability associated to the process deﬁned by the boundary state ψ. This is described in detail, for instance, in the book [21].

How is W deﬁned? As pointed out by Eugenio Bianchi in his Nice lectures [2], the form of W is largely determined by general principles: Feynman’s superposition principle, locality, diﬀeomorphism invariance, crossing symmetry, and local Lorentz invariance.
...
==endquote==

If you'd like to examine this in context, it is on page 7 of the paper. One thing we could do in this thread is to think about each of these general principles and the heuristic way that the quantum gravity transition amplitudes arise from them.

Last edited: Aug 7, 2010
2. Aug 7, 2010

### marcus

"Nice" refers to the city--Bianchi gave a talk at the university there from which part of this drawn. But the approach is also nice in the usual sense and one nice thing is the simplicity of the mathematics.

You imagine a geometric process occurring in a 4D region and you represent it by its boundary state 3D geometry, call that psi.

Psi ψ expresses what we can measure and detect about the initial and final geometry and whatever is relevant and accessible surrounding the 4D region/process. The box surrounding the cat, so to speak.

ψ lives in a hilbertspace H, and there is a linear functional W defined on that hilbertspace which assigns to every 3D boundary state psi a number x+iy in the complex plane which is its amplitude. In mathematics, things rarely get any simpler or nicer than a linear functional on a hilbertspace. And the probability of the process, or the transition from initial to final, is just that number times its complex conjugate x-iy, the square of the absolute value of the number.

Now there's about a page of explanation (part of page 7 and the rest on page 8) about how the transition amplitude, this functional W, arises from or is based on those five principles.
You might wish to ask questions about them that we can try to answer. I think the explanation is actually pretty clear, although quite brief.

3. Aug 7, 2010

### marcus

Some quantum foundations links. Robert Oekl and general boundary formulation of QFT.
http://pirsa.org/09010002/
http://www.matmor.unam.mx/~robert/research.html
http://www.matmor.unam.mx/~robert/index.html
It is possible that Oekl was the first one to present the general boundary formulation and that
Rovelli has adapted it to serve as foundation for Lqg. I remember seeing a paper or papers on this by Oekl back around 2003, 2004 when he was postdoc at Marseille.

I want to try to make this an exposition thread. Trying to explicate what Rovelli's idea is here, rather than debating or contrasting with one's own ideas. I think it is interesting to try to understand in its own right.

Perhaps the most interesting concept is that "there is nothing between events". Einstein proposed this idea---that only events like the collision of two particles have physical meaning. Without being marked by some "elementary process" like an interaction, points have no objective existence.

In a fundamental theory, what does not exist is not represented mathematically. If a point is not marked by the occurrence of some elementary process then its mathematical simulacrum does not belong in a barebones theory. Yes I know this is vague (what is an "event" what is an "elementary process"?) but I am trying to get at the intuitive content of what the article says.

The impact of the diff-invariance principle here is that there is really only vertices. There is a web of geometric relations among these vertices (which stand for elementary geometric processes: chunks of measurable volume dressed in flakes of measurable area.) There is a web of relations between these but the web is not made of anything, it is only some information which we experience. The vertices are, so to speak, the stuff.
There is nothing between the vertices.

All we know about a 4D region is the 3D spin network quantum state describing the boundary. What goes on inside this boundary, in the 4D region, is a superposition of a vast number of possibilities (like Feynman paths) whose amplitudes we can add up. But that is not a picture of reality, it only shows how we plan to calculate the total transition amplitude! We do not say how nature is, inside the spacetime region. All we want is to be able to calculate the transition amplitude corresponding to the boundary.

Last edited: Aug 7, 2010
4. Aug 7, 2010

### marcus

The fourth principle (given as the basis for his version of QG dynamics) can be stated in just a few words. So I will simply quote from page 8:

==quote==

4. Crossing symmetry. It is a well know property of standard QFT that the vertex amplitude does not depend on which states are considered as “in” and which are considered as “out”. Assume the same is true in gravity.

==endquote==

You have probably seen "starfish diagrams" depicting a spinfoam vertex. They are like freeway interchanges where multilane highways come together. If a "chunk" of geometry--that is a vertex--is adjacent to 5 other vertices, then when you diagram it it will look like an interchange where 5 multilane highways come together. This is very figurative, don't take it seriously The flow of traffic at the interchange could represent volume and the traffic along closed loops could represent area. The metaphor doesn't matter, my point is that the vertex has some way of representing volume and area (the geometric essentials).

I am trying to say what crossing symmetry is about.

The important thing is they have some way of packaging the essential geometric information (about what different observers will see when they measure volumes and areas) in the form of a chunk, which is adjacent to other chunks. The dynamic changes in geometry correspond to the creation-annihilation of these chunks according to "moves" or "rules" which have certain amplitudes.
If you like the idea of "pachner moves" (googlable) then you might say the whole assemblage of these chunks "pachnerates". This made-up word is analogous to saying "vibrates" or "jitters" but here we are talking about the rudimentary basic geometry jittering, not weights on springs.

A vertex where 5 roads meet can tell different stories depending on which you say are in-roads and out-roads.

If one road comes in and 4 go out, then it says that 3 new chunks were created. If 3 go in and 2 go out, the story says that one chunk was annihilated.

Crossing symmetry requires that the amplitude of the vertex should not depend on how you assign the ins and outs.

So a "fundamental process" that creates 3 new vertices (the 1 --> 4 move) HAS THE SAME AMPLITUDE as the reverse process that eliminates 3 vertices (the 4 --> 1 move).

That general "crossing symmetry" principle restricts what kind of amplitudes we can have. It illustrates the idea that the geometric transition amplitudes are based on these 5 principles. In effect, the 5 principles (superposition, locality, diff-invariance, crossing, Lorentz-invariance) determine the form of the theory.

Last edited: Aug 8, 2010
5. Aug 7, 2010

### marcus

It's interesting how one should think about the 5th principle---how Lorentz invariance affects the amplitudes. It plays a key role in the general boundary setup (that may possibly be Oekl's contribution).

When you first look at the G.B. setup you might spot a flaw, or mismatch. The whole dynamics of the theory depends on defining this functional W on a SU(2) spin network Hilbert space.

That's because the important thing is the boundary ψ, the "bag" surrounding the 4D region of evolving geometry. The bag contains the information about initial and final geometries and what the observer can observe.

$$<W|\psi>$$

But we define W by a sum over all histories σ that fit inside that boundary and those histories are SL(2,C) spin foams! As in equation (43):

$$<W|\psi> = \sum_\sigma W(\sigma)$$

Now we have to get inside the bag, in effect, and define W not just on spin network states but on spin foams σ ! Extending the definition of W involves breaking the spinfoam down into individual vertices, each surrounded by a small SL(2,C) spin network. To make the construction go through, an SU(2) hilbertspace has to be injected into an SL(2,C) hilbertspace, or to put it another way, we have to map SU(2) group field theory states into an SL(2,C) context. The G.B. formalism requires that.

Fortunately the Peter-Weyl theorem provides each hilbert with a standard basis composed of irreducible representations and the mapping can be done in a natural way by mapping spins j into pairs of numbers (γj, j)---mapping basis elements into basis elements. Gamma γ is the Immirzi number and gamma times j is a positive real number instead of simply a half-integer. Such pairs (a positive real and a half integer) specify an irreducible rep of SL(2,C)

This allows one to define the map fγ that Rovelli talks about at the top of page 7, and defines in equation (41). I've left some loose ends which I'll have to tie down later.

Last edited: Aug 8, 2010
6. Aug 8, 2010

### marcus

I should recapitulate and give links to a couple of Bianchi et al papers which help one understand Rovelli's 1004.1780

What section III of Rovelli's April paper does is base the LQG dynamics on 5 principles.
The kinematics has already been given a kind of Group Field Theory formulation using L2(group manifold, haar measure) spaces to define graph hilberts.
The overall approach follows the General Boundary plan described in work by Oekl.
The G.B. approach is almost certain to be right IMHO. It is how Rovelli managed to derive the LQG graviton propagator in 2006, triggering all the development of new LQG which we are seeing.
The ideas of G.F.T. and of G.B. are firmly in place---the theory is not going to back away from either of them. But that is just a beginning and there is a lot more to be done. So what we are looking at now is how to found the dynamics on these principles:

1. Feynman-style superposition principle
This is simply how the General Boundary plan is implemented: "Following Feynman we expect that the [transition] amplitude...can be expanded in a sum over 'histories of states' where W(σ) is an amplitude associated to an appropriate sequence of states σ."

The amplitude of the bag state is the sum of the stories told inside the bag. That is where this comes from, that we saw before:
$$<W|\psi> = \sum_\sigma W(\sigma)$$

2. Locality
"...Let us, therefore, focus ﬁrst on the amplitude Wv of a single elementary process. This will be interpreted as an elementary vertex, in the same sense in which the QED vertex is the elementary dynamical process that gives an amplitude to the boundary Hilbert space of two electrons and one photon."

3. The "nothing between events" message of diffeomorphism invariance.
In and of themselves, devoid of any occurrence, points of spacetime have no physical meaning---no objective reality. Therefore we do not represent them mathematically in a fundamental theory. There is no manifold. A vertex represents a fundamental process that occurred (eg creation/annihilation of some volume/area). There is nothing "in between". We study a web of geometric relationships among elementary events.

4. Crossing symmetry
This was already discussed a couple of posts back.

5. Lorentz invariance
"Since classical general relativity has a local Lorentz invariance, we expect the individual spinfoam vertex to be Lorentz invariant in an appropriate sense. Since the Hilbert space HΓ deﬁned above has no hint of SL(2,C) action, there should be a map from it to... "

I already discussed this a couple of posts back when talking about the map fγ from the SU(2) to the SL(2,C) states.
================

There are a number of people who are important in the development of LQG after 2008, when it underwent a transformation. We are all aware of Rovelli's work---indeed much of what we see taking shape actually follows some philosophy and general ideas he helped to formulate in the 1990s, including group field theory (and much else besides). But other key contributors to the recent development may not be so familiar. For example, to get an idea of who Eugenio Bianchi is, you can look at his shelf of favorite books, which he has kindly posted here for us:
The profile says he is currently postdoc at Marseille.
http://network.nature.com/profile/eugeniobianchi

Last edited: Aug 8, 2010
7. Aug 8, 2010

### marcus

Earlier I mentioned two papers by Bianchi et al (Eugenio Bianchi, Elena Magliaro, Claudio Perini). They are a great help in understanding the condensed overview by Rovelli that is being discussed in this thread.
http://arxiv.org/abs/0912.4054 Coherent spin-networks
http://arxiv.org/abs/1004.4550 Spinfoams in holomorphic representation
Bianchi et al do not just cover holomorphic spinfoams. They describe several other equivalent spinfoam representations and how they relate the one to the other.

They spell out additional detail about the definition of the map fγ which plays a key role in Lqg dynamics. The easiest way to see the role it plays is to look in Rovelli's April Lqg status report ( http://arxiv.org/abs/1004.1780 ) at the start of section III A, "The LQG vertex."

You will see that the whole dynamics revolves around this remarkably simple-looking equation (45), involving that map. The transition amplitude Wv at spinfoam vertex v is given by:

<Wv|ψ> = (fγψ)(1)

Here 1 is a primitive spin network with links labeled by the identity of the group SL(2,C), surrounding the vertex v.

Transition amplitudes for larger arbitrarily complicated cases are built up from applications of this single vertex case, given in equation (45). For more detail on that, there is Rovelli's page 8, but also see page 5 of the Bianchi et al April paper, equation (35) and following.

Last edited: Aug 8, 2010
8. Aug 9, 2010

### marcus

This treatment of LQG dynamics is new and is based on a recent development which Rovelli summarizes on pages 4 and 5 of the paper (1004.1780).
Several independent ideas of coherent, holomorphic, semiclassical states have converged.
His main focus is on the holomorphic coherent states presented in the Bianchi-Magliaro-Perini paper (0912.4054).
Their description is more leisurely and spread out. Rovelli's is highly condensed. So it is probably a good idea to look at how Bianchi et al tell the story.

==quote from page 2 of "Coherent spin-networks" http://arxiv.org/abs/0912.4054 ==
The coeﬃcients cj of the superposition over spins are given by a Gaussian times a phase as originally proposed by Rovelli in [8] ...
...
Such proposal is motivated by the need of having a state peaked both on the area and on the extrinsic angle. The dispersion is chosen to be given by σ0 ≈ (j0)k (with 0 < k < 2) so that, in the large j0 limit, both variables have vanishing relative dispersions (as explained in [9])...
While the states discussed above have good semiclassical properties and a clear geometrical interpretation, ﬁnding a better top-down derivation of the coeﬃcients (1) is strongly desirable. This is one of the objectives of this paper.

On the other hand, within the canonical framework, Thiemann and collaborators have strongly advocated the use of complexiﬁer coherent states [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Such states are labeled by a graph Γ and by an assignment of a SL(2,C) group element to each of its links. Their peakedness properties have been studied in detail [21, 27]. However the geometric interpretation of the SL(2,C) labels and the relation with semiclassical states used in Spin Foams has largely remained unexplored. Exploring these aspects is the other objective of this paper.

Surprisingly, the two goals discussed above turn out to be strictly related. In this paper we present a proposal of coherent spin-network states : the proposal is to consider the gauge invariant projection of a product over links of Hall’s heat-kernels for the cotangent bundle of SU(2) [30, 31]. The labels of the state are the ones used in Spin Foams: two normals, a spin and an angle for each link of the graph. This set of labels can be written as an element of SL(2,C) per link of the graph. Therefore, these states coincide with Thiemann’s coherent states with the area operator chosen as complexiﬁer, the SL(2,C) labels written in terms of the phase space variables (j0e , ξe, ne, n'e ) and the heat-kernel time given as a function of j0e.

We show that, for large j0e, coherent spin-networks reduce to the semiclassical states used in the spin-foam framework. In particular we ﬁnd that they reproduce a superposition over spins of spin-networks with nodes labeled by Livine-Speziale coherent intertwiners and coeﬃcients cj given by a Gaussian times a phase as originally proposed by Rovelli. This provides a clear interpretation of the geometry these states are peaked on.
==endquote==

So the idea involves replacing the earlier scheme of labeling network links by numbers. Instead of numbers one is going to label the links with group elements. The labels are elements of SL(2,C). There are known ways to break an element of SL(2,C) down and get the numbers you want back, but the new labeling seems conceptually elegant. Plus it unifies a bunch of different proposals by people like Thiemann, Livine, Speziale...quite a few others. Proposals which were motivated by different goals and considerations---so by unifying them, the authors seem to be achieving several different objectives in one move.

And also what they come up with here provides the way that the LQG dynamics is handled in Rovelli's April status report paper. So a degree of economy and order is achieved.

Last edited: Aug 9, 2010
9. Aug 10, 2010

### marcus

I guess if you follow LQG you may have noticed a kind of breakthrough in 0909.0939
Lewandowski et al referring to pages 10,11 (particularly the figures) and where it says in the conclusions:
==quote==
The most important result is the characterization of a vertex of a generalized spin-foam. The
structure of each vertex can be completely encoded in a spin-network induced locally on the boundary of the vertex neighbourhood. The spin-network is used for the natural generalization of the vertex amplitude...
==endquote==

A vertex in a spinfoam has a naturally defined immediate neighborhood and a natural spin-network on the surface, bounding that neighborhood.
The new formulation of spinfoams---how to calculate the vertex amplitude---Lewandowski et al found that it all boils down to simply evaluating the local boundary spin-network.

What Rovelli is telling us in equation (45) is how simple that local boundary spin-network is to calculate. For instance, looking back to post #7 where it gives equation (45)
<Wv|ψ> = (fγψ)(1)

Here v is a spinfoam vertex that you want to calculate the amplitude for. ψ is the spin-network state on the surface of the immediate neighborhood of v. This does not have to be any specific type of network. Let's suppose it is a complete graph with 5 vertices, so it has 10 edges, just for concreteness. Then equation (45) says

<Wv|ψ> = (fγψ)(1,1,1,1,1,1,1,1,1,1)

Here each "1" stands for the identity group element of SL(2,C)
and fγψ is a complex valued square-integrable function defined on 10-tuples of elements of SL(2,C). So it is exactly what is called for to give a number.

fγ essentially gives a map from functions on one group manifold, SU(2)10, to functions on the other group manifold, SL(2,C)10.

Equation (45) gives a concise economical way to say this without all the superfluous notation.

If anyone wants to refer to Lewandowski et al
http://arxiv.org/abs/0909.0939 "Spin-Foams for All Loop Quantum Gravity"
The stress is on the All because they generalize the new 2008 spinfoam vertex, removing restrictions on spinfoams covered by the new definitions. They simplified and broadened the calculation of vertex amplitudes.

Last edited: Aug 10, 2010
10. Aug 10, 2010

### marcus

Pictures can help and Rovelli has "Fig.2" on page 8 which shows how a spinfoam vertex v can relate to the immediate neighborhood boundary spin-network psi.
This corresponds to the pictures on page 11 of Lewandowski et al which show a bit more and spell out more detail.

In Fig.2 it is 3D gravity and the spin-network is the complete graph on 4 points. To surround the vertex it has to have at least 4 points. Also the network is pulled away from the vertex which it would normally surround, so you can see what is happening. Think of Fig.2 as made of stretchy material. Here is what Rovelli says about Fig.2:

==quote http://arxiv.org/abs/1004.1780 ==
Given a spin network state
|ψ⟩ = |Γ, jl , vn⟩, we can visualize the elementary process that has ψ has boundary state as a single vertex (a point), directly connected by edges (lines) to the nodes of Γ and by faces (surfaces) to the links of Γ. See ﬁgure 2.

FIG. 2: Graphical representation of the elementary vertex, for a boundary spin network with Γ formed by the complete graph with 4 nodes (a tetrahedron).

The amplitude of this elementary process will be a function Wv (jl, vn). This function determines the theory.

...
...

Quite astonishingly, the simple and natural vertex amplitude (45) seems to yield the Einstein equations in the large distance classical limit, as I will argue below. A natural group structure based on SU(2) ⊂ SL(2,C) appears ... to code the Einstein equations.
==endquote==

Last edited: Aug 10, 2010
11. Aug 12, 2010

### marcus

This is just my interpretation. If you read the "new Lqg" survey/status report you might find a different message to take away (and I'd be interested to hear it.)

My take is that the main thing it says is "the manifold is gauge".

The spacetime manifold is not physical reality, physical reality corresponds to MEASUREMENT. What differerent observers can observe and detect and measure. That applies to geometry as well as to QED 'electrons' and 'photons'.

The spacetime manifold (invented by Riemann circa 1850) is merely an interpolation method.
Like the 'trajectory' of a 'particle'. We make a finite number of measurements and we run a curve thru, following pre-established conventions.

A manifold is a bunch of conventions associated with excellent versatile methods of interpolation, but we can't say there is one. We can't claim it has objective physical existence. That would require making an uncountable infinity of measurements around every supposed 'point' ---between every pair of points determining the existence of an infinity of intermediate points. An objective continuum is too much to postulate. So we don't.

In this approach the focus is on physical observables---geometric etc. measurement.

So if you look at the development in http://arxiv.org/abs/1780 [Broken], which is now the standard treatment (already in other people's papers for months fait accomplis. I remember someone asking me "isn't it too early to say if the others will go along?" They already went along before It is a review paper, a status report.)
So if you look at the development in the April paper, it is all combinatorial. There is no spacetime manifold.

Last edited by a moderator: May 4, 2017
12. Aug 13, 2010

### marcus

Recently there was a lively protracted discussion about the meaning of diff-invariance in the course of which I quoted that saying of Einstein to the effect that the principle of diff-invariance "deprives space and time of the last remnant of objective physical reality."

In other words, points in the manifold are gauge. They have no objective identity unless of course they are marked by some process, interaction or event----the example often given is a collision, or where two world-lines cross.

Someone commented that Einstein could not possibly have been right, he must have been confused. This may have been a joke, in any case, given the noisy environment, I was not about to argue. But it is an important issue, and Einstein's point has, i believe, been sustained. Whatever else, he was not confused about that one or at least I've seen no evidence that he was.

13. Aug 13, 2010

### suprised

.. well locality seems already very wrong

14. Aug 13, 2010

### Fra

What does this refer to?

/Fredrik

15. Aug 13, 2010

### atyy

Well, that was me and I was not joking. Einstein was right despite being confused - that's the biggest piece of evidence in Rovelli's favour - he could be right despite being wrong.

Actually, not all approaches related to LQG value locality - if you look at Oriti and Rivasseau's expositions of group field theory, nonlocality is stressed.

16. Aug 13, 2010

### suprised

Well one of the lessons of quantum gravity of the recent years is that it is not a local theory, but a holographic one. One way to see this is that the number of degrees of freedom in a given volume goes with the area of the boundary of the volume, not with the volume. People who had been trying to impose the usual rules of local QFT had always failed, for good reasons.

Unfortunately, this and other issues are continued to be confused by uninformed people, doing more harm than good.

For an idea, read eg the introduction in: http://arxiv.org/pdf/hep-th/0203101

17. Aug 13, 2010

### marcus

But surprised, all you do is cite Bousso's article on The Holographic Principle!

I know of no evidence that Lqg is in conflict with the holographic principle. Do you?

BTW I wouldn't claim this proves anything, but two of the top people doing Lqg research are Ashtekar and Freidel. Ashtekar has extended the Bousso covariant entropy bound, using LQG, to cover a case where Bousso's proof broke down.

Freidel may have contributed to proving the AdS/CFT conjecture by giving a construction by which the bulk can be reconstructed (computed) from the boundary information. I'm not sure of the status of this because for whatever reason the paper has not been published (although it has been cited in a major review of the subject by Hans Kastrup.)

Those are just straws in the wind, I expect others will occur to me (just saw your post). I see no sign of conflict. Perhaps others think they do, and will offer some links. I will get the
Ashtekar and Freidel references just in case you are curious.

18. Aug 13, 2010

### Fra

Ok I see what you mean, thanks.

But I think there are alot about the nature of locality and the origin of these information bounds I don't think anyone understands properly.

As I see it, there is a big different between thinking in a realist sense of the degrees of freeom in the objective sense through a communication channel, and to just in a non-realist sense conclude that there is a limit to the inferrable degrees of freedom.

Locality in the sense of only local information influencing local actions, or that the rational action only depends on available information I see no reason to reject.

Edit: Locality with respect to space measures, is something whose statement might not even make much sense if you think space is emergent. I think the more general definition of locality refers not to distance measures in space, but to information divergences generally. So locality seems obvious. It doesn't seem possible to make observation of non-locality. Entanglement of QM is not something I would call non-locality. It's just correlation, not causation. The conclusion of any distributed acquired data is still made locally.

Instead I reject much more strucutral realism. I do not see an objective notion of information to start with, this makes the interpretation of the information bounds much more subtle.

/Fredrik

Last edited: Aug 13, 2010
19. Aug 13, 2010

### suprised

I referred to your claim that locality would be a principle of QG - see the title of the thread. Kinda misleading...

20. Aug 13, 2010

### marcus

Surprised, Why don't you read Rovelli's article and see exactly what he means by each of the five principles that he lists, before you start jumping to conclusions?
Don't go off half-cocked, so to speak. Look on page 7 of the article, where he begins the discussion of locality. And if you want to comment, please read my posts.

Here are the references I mentioned earlier:
http://arxiv.org/abs/0805.3511
The covariant entropy bound and loop quantum cosmology
Abhay Ashtekar, Edward Wilson-Ewing
15 pages, 3 figures
(Submitted on 22 May 2008)
"We examine Bousso's covariant entropy bound conjecture in the context of radiation filled, spatially flat, Friedmann-Robertson-Walker models. The bound is violated near the big bang. However, the hope has been that quantum gravity effects would intervene and protect it. Loop quantum cosmology provides a near ideal setting for investigating this issue. For, on the one hand, quantum geometry effects resolve the singularity and, on the other hand, the wave function is sharply peaked at a quantum corrected but smooth geometry which can supply the structure needed to test the bound. We find that the bound is respected. We suggest that the bound need not be an essential ingredient for a quantum gravity theory but may emerge from it under suitable circumstances."

http://arxiv.org/abs/0804.0632